The closest thing I could find from Nintendo is the IP Requests and Game Submission Ideas:
We appreciate the interest in Nintendo and in all our video game products. To us, it represents a great sign of success and recognition of the Nintendo brand.
"Nintendo" is a registered trademark of Nintendo of America Inc. Nintendo owns extensive intellectual property rights in all of its products, including video game systems, game titles, characters, game software, graphics, artwork, and screen shots. Nintendo also retains rights in content on Nintendo's web sites, including articles, artwork, screen shots and other files. Trademarks and copyrights for third-party games and characters are owned by the companies that market or license those products.
While we are grateful for all the requests for permission to use Nintendo properties, we are not able to grant such requests. We receive thousands of requests and we do not have adequate staffing to review them all. Therefore, our general policy is to decline requests for permission for the use of Nintendo properties.
Although we are not able to grant permission, use of Nintendo's properties without formal permission by Nintendo may still be allowed under the relevant laws of the particular jurisdiction involved. Thus, we encourage you to seek your own legal counsel if you have any questions about whether your particular proposed use is permitted without Nintendo's authorization. Nintendo cannot provide legal advice.
We appreciate your support.
So it appears that this challenge idea would not be legal. However, it seems that the page is talking about spinoff games, not programming contests.
Would such a challenge be legal?
From what I understand, it would be legal if there is enough difference. What is enough difference? If I simply changed every Pokemon name, sprite, move, etc, but left the formulas / game mechanics the same, would this be different enough?
This led me to think about some other challenges. My most recent challenge, May the 4th be with you!, is clearly based off of Star Wars. Star Wars has its own Copyright and Trademarks, so should I have looked at those before posting that challenge? Is it even legal?